Problem: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 5x - 55$, and $ m \angle ABC = 7x - 83$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 83} + {5x - 55} = {90}$ Combine like terms: $ 12x - 138 = 90$ Add $138$ to both sides: $ 12x = 228$ Divide both sides by $12$ to find $x$ $ x = 19$ Substitute $19$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 7({19}) - 83$ Simplify: $ {m\angle ABC = 133 - 83}$ So ${m\angle ABC = 50}$.